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Register a new userQuantitative mappings between symmetry and topology in solids
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The study of spatial symmetries was accomplished during the last century, and had greatly improved our understanding of the properties of solids. Nowadays, the symmetry data of any crystal can be readily extracted from standard first-principles calculation. On the other hand, the topological data (topological invariants), the defining quantities of nontrivial topological states, are in general considerably difficult to obtain, and this difficulty has critically slowed down the search for topological materials. Here, we provide explicit and exhaustive mappings from symmetry data to topological data for arbitrary gapped band structure in the presence of time-reversal symmetry and any one of the 230 space groups. The mappings are completed using the theoretical tools of layer construction and symmetry-based indicators. With these results, finding topological invariants in any given gapped band structure reduces to a simple search in the mapping tables provided.
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The study of topological bandstructures is an active area of research in condensed matter physics and beyond. Here, we combine recent progress in this field with developments in machine-learning, another rising topic of interest. Specifically, we introduce an unsupervised machine-learning approach that searches for and retrieves paths of adiabatic deformations between Hamiltonians, thereby clustering them according to their topological properties. The algorithm is general as it does not rely on a specific parameterization of the Hamiltonian and is readily applicable to any symmetry class. We demonstrate the approach using several different models in both one and two spatial dimensions and for different symmetry classes with and without crystalline symmetries. Accordingly, it is also shown how trivial and topological phases can be diagnosed upon comparing with a generally designated set of trivial atomic insulators.
We extend the notion of fragile topology to periodically-driven systems. We demonstrate driving-induced fragile topology in two different models, namely, the Floquet honeycomb model and the Floquet $pi$-flux square-lattice model. In both cases, we discover a rich phase diagram that includes Floquet fragile topological phases protected by crystalline rotation or mirror symmetries, Floquet Chern insulators, and trivial atomic phases, with distinct boundary features. Remarkably, the transitions between different phases can be feasibly achieved by simply tuning the driving amplitudes, which is a unique feature of driving-enabled topological phenomena. Moreover, corner-localized fractional charges are identified as a ``smoking-gun signal of fragile topology in our systems. Our work paves the way for studying and realizing fragile topology in Floquet systems.
We show that the Wannier obstruction and the fragile topology of the nearly flat bands in twisted bilayer graphene at magic angle are manifestations of the nontrivial topology of two-dimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of two dimensional real fermions in systems with space-time inversion $I_{ST}$ symmetry. The Euler class is an integer topological invariant classifying real two band systems. We show that a two-band system with a nonzero Euler class cannot have an $I_{ST}$-symmetric Wannier representation. Moreover, a two-band system with the Euler class $e_{2}$ has band crossing points whose total winding number is equal to $-2e_2$. Thus the conventional Nielsen-Ninomiya theorem fails in systems with a nonzero Euler class. We propose that the topological phase transition between two insulators carrying distinct Euler classes can be described in terms of the pair creation and annihilation of vortices accompanied by winding number changes across Dirac strings. When the number of bands is bigger than two, there is a $Z_{2}$ topological invariant classifying the band topology, that is, the second Stiefel Whitney class ($w_2$). Two bands with an even (odd) Euler class turn into a system with $w_2=0$ ($w_2=1$) when additional trivial bands are added. Although the nontrivial second Stiefel-Whitney class remains robust against adding trivial bands, it does not impose a Wannier obstruction when the number of bands is bigger than two. However, when the resulting multi-band system with the nontrivial second Stiefel-Whitney class is supplemented by additional chiral symmetry, a nontrivial second-order topology and the associated corner charges are guaranteed.
When the motion of electrons is restricted to a plane under a perpendicular magnetic field B, a variety of quantum phases emerge at low temperatures whose properties are dictated by the Coulomb interaction and its interplay with disorder. At very strong B, the sequence of fractional quantum Hall (FQH) liquid phases terminates in an insulating phase, which is widely believed to be due to the solidification of electrons into domains possessing Wigner crystal (WC) order. The existence of such WC domains is signaled by the emergence of microwave pinning-mode resonances, which reflect the mechanical properties characteristic of a solid. However, the most direct manifestation of the broken translational symmetry accompanying the solidification - the spatial modulation of particles probability amplitude - has not been observed yet. Here, we demonstrate that nuclear magnetic resonance (NMR) provides a direct probe of the density topography of electron solids in the integer and fractional quantum Hall regimes. The data uncover quantum and thermal fluctuation of lattice electrons resolved on the nanometre scale. Our results pave the way to studies of other exotic phases with non-trivial spatial spin/charge order.
We analyze the possible interaction-induced superconducting instabilities in noncentrosymmetric systems based on symmetries of the normal state. It is proven that pure electron-phonon coupling will always lead to a fully gapped superconductor that does not break time-reversal symmetry and is topologically trivial. We show that topologically nontrivial behavior can be induced by magnetic doping without gapping out the resulting Kramers pair of Majorana edge modes. In case of superconductivity arising from the particle-hole fluctuations associated with a competing instability, the properties of the condensate crucially depend on the time-reversal behavior of the order parameter of the competing instability. When the order parameter preserves time-reversal symmetry, we obtain exactly the same properties as in case of phonons. If it is odd under time-reversal, the Cooper channel of the interaction will be fully repulsive leading to sign changes of the gap and making spontaneous time-reversal symmetry breaking possible. To discuss topological properties, we focus on fully gapped time-reversal symmetric superconductors and derive constraints on possible pairing states that yield necessary conditions for the emergence of topologically nontrivial superconductivity. These conditions might serve as a tool in the search for topological superconductors. We also discuss implications for oxides heterostructures and single-layer FeSe.
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